This chapter addresses the issue of gauge fixing necessary for quantizing Abelian gauge fields, specifically the photon field in Quantum Electrodynamics (QED), using the path integral formalism. It reviews the local gauge invariance of the free electromagnetic Lagrangian, which leads to a division by an infinite volume in the path integral space, rendering the generating functional ill-defined. The chapter introduces the solution using the Fadeev-Popov procedure, which inserts a term into the path integral to fix the gauge. This leads to the Lorenz gauge fixing term and the resulting photon propagator. Finally, the full path integral for interacting QED is written down, incorporating the gauge-fixing term, the Dirac field path integral, and the Feynman rules needed for calculating invariant amplitudes. The chapter concludes by demonstrating the importance of the Ward identity in preserving the physical requirements of the theory despite the introduction of unphysical gauge-dependent terms.

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Path Integrals for Abelian Gauge Fields

  • Michael Strickland

摘要

This chapter addresses the issue of gauge fixing necessary for quantizing Abelian gauge fields, specifically the photon field in Quantum Electrodynamics (QED), using the path integral formalism. It reviews the local gauge invariance of the free electromagnetic Lagrangian, which leads to a division by an infinite volume in the path integral space, rendering the generating functional ill-defined. The chapter introduces the solution using the Fadeev-Popov procedure, which inserts a term into the path integral to fix the gauge. This leads to the Lorenz gauge fixing term and the resulting photon propagator. Finally, the full path integral for interacting QED is written down, incorporating the gauge-fixing term, the Dirac field path integral, and the Feynman rules needed for calculating invariant amplitudes. The chapter concludes by demonstrating the importance of the Ward identity in preserving the physical requirements of the theory despite the introduction of unphysical gauge-dependent terms.